In this paper, I present an interpretation of the use of constructions in both the problems and theorems of Elements I–VI, in light of the concept of given as developed in the Data, that makes a distinction between the way that constructions are used in problems, problem-constructions, and the way that they are used in theorems and in the proofs of problems, proof-constructions. I begin by showing that the general structure of a problem is slightly different from that stated by Proclus in his commentary on the Elements. I then give a reading of all five postulates, Elem. I.post.1–5, in terms of the concept of given. This is followed by a detailed exhibition of the syntax of problem-constructions, which shows that these are not practical instructions for using a straightedge and compass, but rather demonstrations of the existence of an effective procedure for introducing geometric objects, which procedure is reducible to operations of the postulates but not directly stated in terms of the postulates. Finally, I argue that theorems and the proofs of problems employ a wider range of constructive and semi- and non-constructive assumptions that those made possible by problems.

中文翻译：

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构造在欧几里得元素 I-VI 中的问题和定理中的应用

在本文中，我根据数据中提出的给定概念，对元素 I-VI 的问题和定理中构造的使用进行了解释，该解释区分了构造在元素 I-VI 中的使用方式。问题，问题构造，以及它们在定理和问题证明中的使用方式，证明构造。我首先表明问题的一般结构与 Proclus 在他对元素的评论中所述的结构略有不同。然后我阅读所有五个假设，Elem。I.post.1-5，就给定的概念而言。接下来是对问题结构语法的详细展示，这表明这些不是使用直尺和指南针的实用说明，而是证明存在用于引入几何对象的有效程序，该程序可简化为假设的操作，但不直接根据假设进行陈述。最后，我认为定理和问题的证明采用了更广泛的建设性、半建设性和非建设性假设，这些假设是由问题实现的。